I actually watched FOTR before I read the books. The movie transfixed me, and I determined to read the books. I haven't been the same since!
I fell in love with the attention to detail, the descriptions of nature, and the classic character-types presented as if mythology had been rewritten in the 20th century.
The books now fly by when I reread them, and I'm currently on my 6 or 7th go around.
They just don't get old.

Favorite Tolkien character:

I'm a huge fan of the Silmarillion, and in that vein, I really dig Thingol: he is after all the forerunner of all the main movers and shakers. From Luthien, Elrond, to Aragorn, he was the beginning of it all. It also doesn't hurt that he is married to Melian of the Maiar.